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Effects of Slip Boundary Conditions on Rayleigh-Bénard Convection

Published online by Cambridge University Press:  05 May 2011

L.-S. Kuo  and
P.-H. Chen
L.-S. Kuo*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
P.-H. Chen*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Graduate student
**Professor, corresponding author
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Abstract

This work studied the Rayleigh-Bénard convection under the first-order slip boundary conditions in both hydrodynamic and thermal fields. The variation principle was applied to find the critical Rayleigh number of instability. The exteneded relations of the critical Rayleigh number (Rc) and the wavenumber (ac) under partially slip boundary conditions were derived. The numerical results showed that both Rc and ac are decreasing with increasing the Knudsen number. The dependence of Rc on the Knudsen number (K) shows that when K≤10−3, the boundary can be considered as nonslip, while K≥10, it can be considered as free boundaries. The maximum change rate occurs when the Knudsen number is around 0.1, indicating that the system would be affected significantly in that range.

Keywords

Slip velocityTemperature jumpTangential momentum accommodation coefficientRayleigh-Bénard convection
Type
Articles
Information
Journal of Mechanics , Volume 25 , Issue 2 , June 2009 , pp. 205 - 212
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2009

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References

1.Maxwell, J., “On Stresses in Rarified Gases Arising from Inequalities of Temperature,” Phil. Trans. R. Soc. Lond., 170, p. 231 (1879).Google Scholar
2.Arkilic, E. B., Breuer, K. S. and Schmidt, M. A., “Mass Flow and Tangential Momentum Accommodation in Silicon Micromachined Channels,” J. Fluid Mech., 437, p. 29 (2001).CrossRefGoogle Scholar
3.Arkilic, E. B., Schmidt, M. A. and Breuer, K. S., “Gaseous Slip Flow in Long Microchannels,” J. Microelectromech. Syst., 6, p. 167 (1997).CrossRefGoogle Scholar
4.Maurer, J., Tabeling, P., Joseph, P. and Willaime, H., “Second-Order Slip Laws in Microchannels for Helium and Nitrogen,” Phys. Fluids, 15, p. 2613 (2003).CrossRefGoogle Scholar
5.Craig, V. S. J., Neto, C. and Williams, D. R. M., “Shear-Dependent Boundary Slip in an Aqueous Newtonian Liquid,” Phys. Rev. Lett., 87, 054504 (2001).CrossRefGoogle Scholar
6.Vinogradova, O. I., “Slippage of Water Over Hydrophobic Surfaces,” Int. J. Miner. Process, 56, p. 31 (1999).CrossRefGoogle Scholar
7.Zhu, Y. X. and Granick, S., “Rate-Dependent Slip of Newtonian Liquid at Smooth Surfaces,” Phys. Rev. Lett., 87, 096105 (2001).Google Scholar
8.Squires, T. M. and Quake, S. R., “Microfluidics: Fluid Physics at the Nanoliter Scale,” Rev. Mod. Phys., 77, p. 977 (2005).Google Scholar
9.Neto, C., Evans, D. R., Bonaccurso, E., Butt, H. J. and Craig, V. S. J., “Boundary Slip in Newtonian Liquids: A Review of Experimental Studies,” Rep. Prog. Phys., 68, p. 2859 (2005).Google Scholar
10.Lorenz, E. N., “Deterministic Nonperiodic Flow,” J. Atmos. Sci., 20, p. 130 (1963).Google Scholar
11.Cross, M. C. and Hohenberg, P. C., “Pattern-Formation Outside of Equilibrium,” Rev. Mod. Phys., 65, p. 851 (1993).Google Scholar
12.Krishnan, M., Ugaz, V. M. and Burns, M. A., “PCR in a Rayleigh-Benard Convection Cell,” Science, 298, p. 793 (2002).CrossRefGoogle Scholar
13.Bodenschatz, E., Pesch, W. and Ahlers, G., “Recent Developments in Rayleigh-Benard Convection,” Annu. Rev. FluidMech., 32, p. 709 (2000).CrossRefGoogle Scholar
14.Drazin, P. and Reid, W. H., Hydrodynamic Stability, 2nd Ed., Cambridge Univ Press (2004).CrossRefGoogle Scholar
15.Karniadakis, G., Beskok, A. and Aluru, N., Microflows and Nanoflows: Fundamentals and Simulation, 1st Ed., Springer (2005).Google Scholar
16.Lo, D. C., Liao, T., Young, D. L. and Gou, M. H., “Velocity-Vorticity Formulation for 2D Natural Convection in an Inclined Cavity by the DQ Method,” J. Mech., 23, p. 261 (2007).CrossRefGoogle Scholar
17.Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover (1981).Google Scholar
18.Manela, A. and Frankel, I., “On the Rayleigh-Benard Problem in the Continuum Limit,” Phys. Fluids, 17, 036101 (2005).Google Scholar
19.Stefanov, S., Roussinov, V. and Cercignani, C., “Rayleigh-Benard Flow of a Rarefied Gas and Its Attractors, I. Convection Regime,” Phys. Fluids, 14, p. 2255 (2002).Google Scholar
20.Stefanov, S., Roussinov, V. and Cercignani, C., “Rayleigh-Benard Flow of a Rarefied Gas and Its Attractors, II. Chaotic and Periodic Convective Regimes,” Phys. Fluids, 14, p. 2270 (2002).Google Scholar
21.Barber, R. W., Sun, Y., Gu, X. J. and Emerson, D. R., “Isothermal Slip Flow Over Curved Surfaces,” Vacuum, 76, p. 73 (2004).CrossRefGoogle Scholar